# Local Minimization as a Selection Criterion

## Abstract

The *Γ*-limit *F* of a sequence \(F_{\varepsilon }\) is often interpreted as a simplified description of the energies \(F_{\varepsilon }\), where unimportant details have been averaged out still keeping the relevant information about minimum problems. As far as global minimization problems are concerned this is ensured by the fundamental theorem of *Γ*-convergence, but this is in general false for local minimization problems. Nevertheless, if some information on the local minima is known, we may use the fidelity of the description of local minimizers as a way to “correct” *Γ*-limits. In order to do that, we first introduce some notions of *equivalence by Γ -convergence*, and then show how to construct simpler equivalent theories as perturbations of the *Γ*-limit *F* in some relevant examples. We will exhibit a sharp-interface theory equivalent to the scalar Ginzburg–Landau theory, and derive Barenblatt theory of fracture as a perturbation of Griffith theory maintaining the pattern of minimizers of Lennard-Jones systems.

## Keywords

Discontinuity Point Local Minimization Problem Strict Local Minimizer Homogeneous Functional Finite Perimeter## References

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